Cody Tipton | Searcharxiv

arXiv2024

$n\text{-}Lie_d$ Operad and its Koszul Dual

We study the operad $n\text{-}Lie_d$, whose algebras are graded $n$-Lie algebras with degree $d$ $n$-arity operations, which were introduced in Nambu mechanics and later studied in the algebraic setting with Filippov. We compute the Koszul dual of $n\text{-}Lie_d$, called $n\text{-}Com_{-d+n-2}$, whose relations are derived from the Specht module $S^{(n,n-1)}$ for a partition $(n,n-1)$ of $2n-1$. The intrinsic connection between these two operads come from the eigenvalues of the sequence of graphs $\{\mathcal{O}_n\}_{n\geq 0}$, called the Odd graphs, whose spectrum is related to the lower triangular sequence $\{\mathcal{E}_{r,n}\}$, called the Catalan triangle.

math.RA↗

arXiv2023

Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {\emph{Haldane bundle dataset}}, which consists of synthetically generated complex line bundles on the $2$-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

cond-mat.mes-hall↗